recovering the water-wave pro le from pressure measurements
TRANSCRIPT
Recovering the water-wave profile from pressure measurements
K. L. Oliveras∗, V. Vasan†, B. Deconinck‡, D. Henderson§§
∗Department of Mathematics, Seattle University, Seattle, WA 98122† ‡ Department of Applied Mathematics, University of Washington, Seattle, WA 98195-2420
§Department of Mathematics, Penn State University, University Park, PA 16802
November 9, 2011
This paper is dedicated to the memory of Joe Hammack,who provided the impetus for this project.
Abstract
A new method is proposed to recover the water-wave surface elevation from pressure dataobtained at the bottom of the fluid. The new method requires the numerical solution of anonlocal nonlinear equation relating the pressure and the surface elevation which is obtainedfrom the Euler formulation of the water-wave problem without approximation. From this newequation, a variety of different asymptotic formulas are derived. The nonlocal equation andthe asymptotic formulas are compared with both numerical data and physical experiments.The solvability properties of the nonlocal equation are rigorously analyzed using the ImplicitFunction Theorem.
1 Introduction
In field experiments, the elevation of a surface water-wave in shallow water is often determined bymeasuring the pressure along the bottom of the fluid, see e.g. [4], [6], [18], [19], [24], [25]. A varietyof approaches are used for this. The two most commonly used are the hydrostatic approximationand the transfer function approach. For the hydrostatic approximation [9, 17],
η(x, t) =P (x,−h, t)
ρg− h, (1)
where g is the acceleration due to gravity, h represents the average depth of the fluid, ρ is the fluiddensity, P (x,−h, t) is the pressure as a function of space x and time t evaluated at the bottom ofthe fluid z = −h, and η(x, t) is the zero-average surface elevation. Throughout, we assume thatall wave motion is one-dimensional with only one horizontal spatial variable x. The hydrostaticapproximation is used, for instance, in open-ocean buoys employed for tsunami detection, see [21].
∗[email protected]†[email protected]‡[email protected]§[email protected]
1
The transfer function approach uses a linear relationship between the Fourier transforms F ofthe dynamical part of the pressure and the elevation of the surface [9, 13, 16, 17]:
F {η(x, t)} (k) = cosh(kh)F {p(x, t)/g} (k), (2)
where p(x, t) = (P (x,−h, t)− ρgh)/ρ is the dynamic (or non-static) part of the pressure P (x, z, t)evaluated at the bottom of the fluid z = −h, scaled by the fluid density ρ. In this relationship, ηand p are regarded as functions of the spatial coordinate x, with parametric dependence on time t.It is equally useful to let t vary for fixed x, as would be appropriate for a time series measurement,which results in extra factors of the wave speed c(k), due to the presence of a temporal instead ofa spatial Fourier transform.
It is well understood that nonlinear effects play a significant role when reconstructing the surfaceelevation for shallow-water waves or for waves in the surf zone (see [5, 6, 25], for instance). Sincenonlinear effects are not captured by the linear transfer function (2), different modifications of(2) have been proposed. One approach is to modify the transfer function to incorporate extraparameters (e.g., multiplicative factors, width scalings) that are tuned to fit data [18, 19, 25]. Aless empirical approach is followed in [5] and [20] where corrections to the transfer function areproposed based on higher-order Stokes expansions. Bishop & Donelan [6] examine the empiricalapproaches and argue that the inclusion of the proposed parameters is not necessary as errors frominaccuracy in instrumentation and analysis are likely to outweigh the benefit of their presence. Wedo not include any of the modified transfer function approaches in our comparisons.
Bishop and Donelan [6] acknowledge that the linear response cannot accurately capture nonlin-ear effects. While both the hydrostatic model and the transfer function approach are accurate onsome scales, they fail to reconstruct the surface elevation accurately in the case of large-amplitudewaves, as might be expected. Errors of 15% or more are common, as is shown and discussed below.In order to address the inaccuracies of the linear models, nonlinear methods are required. With theexception of recent work by Escher & Schlurmann [13] and Constantin & Strauss [8], few nonlinearresults are found in the literature. Escher & Schlurmann [13] provide a consistent derivation of (2)and offer some thoughts about the impact of nonlinear effects. Starting from a traveling wave as-sumption, Constantin & Strauss [8] obtain different properties and bounds relating the pressure andsurface elevation. However, they do not present a reconstruction method to accurately determineone function in terms of the other.
One way to obtain an improved pressure-to-surface elevation map is to use perturbation methodsto determine nonlinear correction terms to (2). Several such approaches are given below, and weinclude them when comparing the different methods. Our main focus, however, is the presentationof a new nonlocal nonlinear relationship between the pressure at the bottom of the fluid, and theelevation of a traveling-wave surface that captures the full nonlinearity of Euler’s Equations. Theadvantage of this approach is that
1. it allows for the surface to be reconstructed numerically from any given pressure data for atraveling wave,
2. it provides an environment for the direct analysis of the relationship between all physicallyrelevant parameters such as depth and wave speed,
3. and it allows for the quick derivation of perturbation expansions such as the ones mentionedabove.
2
x
z
z = η(x, t)
D
z = −h
z = 0
P (x,−h, t)
Figure 1: The fluid domain D for the water wave problem. An idealized pressure sensor is indicatedat the bottom. In our calculations the pressure measurement is assumed to be a point measurement.
4. Although our approach is formally limited to traveling waves, it can be applied with greatsuccess to more general wave profiles that are not merely traveling. This is illustrated anddiscussed below.
In what follows, we derive these nonlocal relations and demonstrate their practicality. Wecompare results from the nonlocal formulation with those from the linear approaches and differentnonlinear perturbative models, using both numerical data for traveling waves in shallow water, andexperimental data obtained at Penn State’s Pritchard Fluid Mechanics Laboratory. We demon-strate the superiority of the nonlocal reconstruction formula for a large range of amplitudes. Inaddition, using the Implicit Function Theorem, we analyze the nonlocal formulation in order todemonstrate its solvability for the surface elevation given the pressure.
2 A nonlocal formula relating pressure and surface elevation
Consider Euler’s equations describing the dynamics of the surface of an ideal fluid in two dimensions(with a one-dimensional surface):
φxx + φzz = 0, (x, z) ∈ D, (3)
φz = 0, z = −h, (4)
ηt + ηxφx = φz, z = η(x, t), (5)
φt +1
2
(φ2x + φ2z
)+ gη = 0, z = η(x, t), (6)
where φ(x, z, t) represents the velocity potential of the fluid with surface elevation η(x, t). As posed,the equations require the solution of Laplace’s equation inside the fluid domain D, see Figure 1.If the problem is posed on the whole line x ∈ R, we require that all quantities approach zero atinfinity. If periodic boundary conditions are used then all quantities at the right end of the fluiddomain are equal to those at the left end. Below we work with the whole line problem, stating onlythe results for the periodic case.
Following [26], let q(x, t) represent the velocity potential at the surface z = η(x, t), so that
q(x, t) = φ(x, η(x, t), t). (7)
Combining the above with equation (5), we have
3
φz = ηt + (qx − φzηx) ηx,
obtained by taking an x-derivative of (7). This allows us to solve directly for φz in terms of η andq:
φz =ηt + ηxqx
1 + η2x. (8)
Using (5) again gives an expression for φx, while taking a t-derivative of (7) leads to an expressionfor φt:
φx =qx − ηxηt
1 + η2x, φt = qt −
ηt (ηt + ηxqx)
1 + η2x. (9)
Substituting these expressions into the dynamic boundary condition (6) we find
qt +1
2q2x + gη − 1
2
(ηt + qxηx)2
1 + η2x= 0, (10)
after some simplification.Next, we restrict to the case of a traveling wave moving with velocity c. We introduce ξ = x−ct,
so that x and t-derivatives become ξ-derivatives, the latter multiplied by−c. The Bernoulli equation(10) becomes a quadratic equation in qξ:
−cqξ +1
2q2ξ + gη − 1
2
η2ξ (qξ − c)21 + η2ξ
= 0. (11)
Solving this quadratic equation, we find
qξ = c±√
(c2 − 2gη)(1 + η2ξ ), (12)
where for c > 0 we choose the − sign, to ensure that the local horizontal velocity is less than thewave speed [8]. Similarly, for c < 0 the + sign should be chosen.
Substituting this result into (8) and (9), we find
φξ = c−√c2 − 2gη
1 + η2ξ, φz = −ηξ
√c2 − 2gη
1 + η2ξ, (13)
where we have chosen c > 0, without loss of generality. This simple calculations allows us to toexpress the gradient of the velocity potential at the surface directly in terms of the surface elevation.
Returning to the original coordinate system (x, z, t), let Q(x, t) = φ(x,−h, t), the velocitypotential at the bottom of the fluid. Inside the fluid, we know that the Bernoulli equation holds:
φt +1
2
(φ2x + φ2z
)+ gz +
P (x, z, t)
ρ= 0, − h ≤ z ≤ η(x, t). (14)
Evaluating this equation at z = −h, we find
Qt +1
2Q2x − gh+
P (x,−h, t)ρ
= 0. (15)
4
Moving to a traveling coordinate frame as before, we obtain a quadratic equation for Qξ. Solvingfor Qξ we find
Qξ = c−√c2 − 2p, (16)
where p(ξ) represents the non-static part of the pressure at the bottom in the traveling coordinateframe, scaled by the fluid density: p(ξ) = P (x−ct,−h, t)/ρ−gh. For consistency with our previouschoice, we work with the − sign again. Next, we connect the information at the surface with thatat the bottom of the fluid.
Within the bulk of the fluid D,φξξ + φzz = 0, (17)
where the boundary conditions given in (13) and (16) must also be satisfied. We can write thesolution of this equation as
φ(ξ, z) =1
2π
∫ ∞−∞
eikξΨ(k) cosh (k (z + h)) dk, (18)
where the boundary condition for φz at z = −h is satisfied. For the boundary condition at thebottom for φξ we find
1
2π
∫ ∞−∞
ikeikξΨ(k) dk = c−√c2 − 2p, (19)
so that
ikΨ(k) = 2πcδ(k)−F{√
c2 − 2p}
(k), (20)
where δ(k) is the Dirac delta function and F denotes the Fourier transform: F{y(ξ)}(k) =∫∞−∞ y(ξ) exp(−ikξ)dξ. Evaluating φξ(ξ, z) at the surface z = η, we have
φξ(ξ, η) =1
2π
∫ ∞−∞
eikξikΨ(k) cosh (k (η + h)) dk
= c− 1
2π
∫ ∞−∞
eikξ cosh (k (η + h))F{√
c2 − 2p}
(k) dk.
Using the boundary conditions given in (13), we find the nonlocal relationship√c2 − 2gη
1 + η2ξ=
1
2π
∫ ∞−∞
eikξ cosh (k (η + h))F{√
c2 − 2p}
(k) dk. (21)
Equation (21) is the main result of this paper. It provides an implicit relationship between thesurface elevation of a localized traveling wave η(x) and the pressure measured at the bottom of thefluid p(ξ). In the rest of this paper we investigate how this relationship may be used to computeη(ξ) if p(ξ) is known, and how different asymptotic formulas may be derived from it.
5
Remarks.
• In order to extend the above to periodic boundary conditions, we use the periodic generaliza-tion of the formulation of Ablowitz, Fokas, & Musslimani AFM, see [11]. Following the stepsoutlined above, this allows for the derivation of a relation between the surface elevation of aperiodic traveling wave and the pressure at the bottom:√
c2 − 2gη
1 + η2ξ=
1
2π
∞∑k=−∞
eikξ cosh (k (η + h)) Pk, (22)
where Pk =∫ 2π0 e−ikξ
√c2 − 2p(ξ) dξ. In what follows, we will use either (21) or (22).
• In the above, we have assumed there exists a smooth solution to the water wave problem(3-6). Given a speed c and a non-hydrostatic pressure profile p as inputs, we aim to solve (21)for η. However, these inputs cannot be independent of each other: indeed arbitrary pressureprofiles will not lead to surface elevations corresponding to solutions of (3-6). One expectsthat for a given speed c, there exists a unique surface elevation η and associated pressureprofile p. In order to back up this intuition, we require another relation between η, p andc. Such a relation is found by taking a derivative with respect to z of (18) and equating theresult to the right-hand side of the second equality in (13). Finally, (20) is used, resulting in
ηx
√c2 − 2gη
1 + η2x=−i2π
∫ ∞−∞
eikx sinh(k(η + h))F{c−
√c2 − 2p
}(k)dk. (23)
The system of equations (21) and (23) may be solved to obtain both η and p, given c. Wewill not pursue this issue further and content ourselves with establishing a map from p to η.
For the purposes of the question considered in this paper, the above is not an issue: we assumethat the given pressure originates from experimental observations and hence corresponds tothe unique solution of (3-6), to the extent that the Euler equations provide an accurate modelfor the water wave problem.
• Obtaining the pressure at the bottom from the surface elevation. An explicitnonlinear relationship for the pressure at the bottom in terms of the surface elevation maybe obtained directly from the approach of Ablowitz, Fokas and Musslimani. Consider therelationship (the one-dimensional version of Equation 1.11 in [1])∫ ∞
−∞eikx±|k|(η+h) (sgn(k)ηt − i qx) dx = −i
∫ ∞−∞
eikxφx(x,−h, t)dx, k ∈ R0. (24)
Changing to a moving frame of reference and substituting the expressions (12,16) into thisrelation one obtains
∫ ∞−∞eikx±|k|(η+h)
(−ic sgn(k)ηx +
(c−
√(c2 − 2gη)(1 + η2x)
))dx =
∫ ∞−∞eikx
(c−
√c2 − 2p
)dx.
(25)
6
The right-hand side is essentially the Fourier transform of the quantity c−√c2 − 2p. Inverting
this transform, one may solve for the pressure at the bottom in terms of the surface elevation.Equation (25) provides an alternative to (21) and can be used in its stead. The formula(25) is advantageous if one wishes to compute the pressure, given the surface elevation. Thepresence of a small-divisor problem (seen by linearizing the left-hand side integrand aboutη = 0) when c is near its shallow-water limit value
√gh indicates that (21) is to be preferred
over (25) to reconstruct the elevation η, given the bottom pressure p.
• In [8], Constantin and Strauss derive various properties of the pressure underneath a travelingwave. It should be possible to re-derive these properties directly from (21) or (22). This isnot pursued further here.
3 Existence and uniqueness of solutions to the nonlinear formula
In this section, we analyze (21). Among other results, using the Implicit Function Theorem, weshow that the nonlocal relation (21) gives rise to a well-defined map from the pressure profile to thesurface elevation: given the pressure profile p at the bottom, (21) defines a unique surface elevationη. In other words, we can expect the asymptotic and numerical methods employed in the nextsections to produce faithful approximations to the true solution.
Define the operator F , parameterized by c ∈ R, by
F (η, p) = c−√c2 − 2gη
1 + η2x− 1
2π
∫ ∞−∞
eikx cosh(k(η + h))F{c−
√c2 − 2p
}(k)dk. (26)
Note that F (η, p) = 0 is equivalent to (21). Using the Implicit Function Theorem, we wish to showthat the equation F (η, p) = 0 has a solution profile η, given sufficiently small pressure p. We havethat F (0, 0) = 0. In order to apply the Implicit Function Theorem, we need to define appropriateBanach spaces for which the operator F is defined. First we seek a suitable space for η. An obviouschoice is η ∈ C1[R,R], i.e. η is a continuously differentiable function which vanishes at infinity.This space is supplied with the usual norm:
‖η‖C1 = supx∈R|η(x)|+ sup
x∈R|η′(x)|. (27)
If ‖η‖C1 < c2/2g then c −√
(c2 − 2gη)/(1 + η2x) represents a continuous function of x. Hencewe are motivated to define the image of F in C[R,R]. Consequently, we wish for∫ ∞
−∞eikx cosh(k(η + h))F
{c−
√c2 − 2p
}(k)dk,
to be a continuous function of x. For finite ‖η‖C1 , this nonlocal term is a continuous function of xif ∫ ∞
−∞cosh(k(‖η‖C1 + h))
∣∣∣F {c−√c2 − 2p}(k)∣∣∣ dk <∞, (28)
and if the integrand of the nonlocal term is a continuous function of x for every k (see Theorem 2.27on pg. 56 of [15]). Let us consider the second condition, namely the continuity of the integrand.Since by assumption η is continuous, the continuity in x of the integrand requires
7
supk
∣∣∣F {c−√c2 − 2p}(k)∣∣∣ <∞. (29)
An application of the Cauchy-Schwarz inequality gives
∣∣∣F {c−√c2 − 2p}
(k)∣∣∣ ≤ (∫ ∞
−∞
1
1 + |x|2dx) 1
2(∫ ∞−∞
(1 + |x|2)∣∣∣c−√c2 − 2p
∣∣∣2 dx) 12
.
Hence, we impose the following condition on the pressure p:∫ ∞−∞
(1 + |x|2)∣∣∣c−√c2 − 2p
∣∣∣2 dx <∞. (30)
Next, we return to the first condition (28). Due to the presence of the hyperbolic cosine,we expect that it is necessary for F{c −
√c2 − 2p}(k) to have sufficient decay for large |k|. Let
M > h+ ‖η‖C1 . Starting from the integral in (28), we apply the Cauchy-Schwarz inequality againto find
∫ ∞−∞
cosh(k(‖η‖C1 + h))e−M |k|eM |k|∣∣∣F {c−√c2 − 2p
}(k)∣∣∣ dk
≤(∫ ∞−∞
(cosh(k(‖η‖C1 + h))e−M |k|
)2dk
) 12(∫ ∞−∞
e2M |k|∣∣∣F {c−√c2 − 2p
}(k)∣∣∣2dk) 1
2
≤ C(∫ ∞−∞
e2M |k|∣∣∣F {c−√c2 − 2p
}(k)∣∣∣2dk) 1
2
,
for some constant C. Thus, if the conditions
∫ ∞−∞
e2M |k|∣∣∣F {c−√c2 − 2p
}(k)∣∣∣2 dk <∞ (31)
and
∫ ∞−∞
(1 + |x|2)∣∣∣c−√c2 − 2p
∣∣∣2 dx <∞ (32)
hold, the function F (η, p)(x) is continuous for η ∈ C1.Having found the conditions (31-32), we now determine an appropriate function space for p so
that they are satisfied. The following theorem due to Paley and Wiener (Theorem 4 pg. 7 of [23])is helpful.
Theorem 3.1 If w(z) (with z = x + iy) is an analytic function in the strip −λ ≤ y ≤ µ whereλ, µ > 0 and ∫ ∞
−∞|w(x+ iy)|2dx <∞, −λ ≤ y ≤ µ,
then there exists a measurable function w(k) such that
8
∫ ∞−∞|w(k)|2e2µkdk <∞,
∫ ∞−∞|w(k)|2e−2λkdk <∞,
and
w(x+ iy) = limA→∞
∫ A
−A
1
2πw(k)eik(x+iy)dk, −λ ≤ y ≤ µ
where the limit is to be understood in the mean-square sense.
In other words, the Fourier transform of w(x) exists and it has decay as specified above. Inparticular, for any M < min{λ, µ}
∫ ∞−∞
e2M |k||w(k)|2dk =
∫ ∞0
e2Mk|w(k)|2dk +
∫ 0
−∞e−2Mk|w(k)|2dk,
=
∫ ∞0
e2Mke−2µke2µk|w(k)|2dk +
∫ 0
−∞e−2Mke2λke−2λk|w(k)|2dk,
≤∫ ∞0
e2µk|w(k)|2dk +
∫ 0
−∞e−2λk|w(k)|2dk <∞.
The theorem implies that a sufficient condition for (31) to hold is that c −√c2 − 2p is an
analytic function of z within a strip of width at least 2M centered around the real axis and thatit is square-integrable along lines parallel to the real axis within this strip. Of course, the presenceof the square root is a hinderance to the analyticity of the function. Consequently, we requirethat |p| < c2/2 everywhere in the strip. This condition also implies the square-integrability of thefunction if p is square-integrable. Indeed, the function
f(z) = c−√c2 − 2z,
is analytic (and hence Lipschitz) in a neighborhood of the origin for which |z| < c2/2. Hence, forall z1, z2 in such a neighborhood of the origin we have
|f(z1)− f(z2)| ≤ C|z1 − z2|, (33)
for some constant C. In particular, since f(0) = 0,
|f(z)| ≤ C|z|, (34)
uniformly for all |z| ≤ δ < c2/2, i.e., the constant C is independent of z. Next, consider thefunction f(p(z)), where p(z) is a function which is analytic and bounded in the strip of width 2M .This implies
|f(p(z))| ≤ C|p(z)|. (35)
As above, the constant C is independent of p(z) and thus of z, provided |p(z)| < c2/2. Thusthe square-integrability of p(z) implies the square-integrability of c −
√c2 − 2p for |p| ≤ δ < c2/2
for every z in the strip. Thus, if p(z) is an analytic function in the strip of width at least 2M ,
9
square-integrable along lines parallel to the real axis, and bounded in the strip, the first condition(31) holds.
Another theorem due to Paley and Wiener (Theorem 2 pg 5 of [23]) allows us to bound |p| interms of the L2-norm.
Theorem 3.2 Let w(z) be analytic in the strip −λ ≤ y ≤ µ with µ, λ > 0 and∫ ∞−∞|w(x+ iy)|2dx <∞, −λ ≤ y ≤ µ, (36)
then for any z in the interior of the region
w(z) =1
2π
∫ ∞−∞
w(x+ iµ)
x+ iµ− z dx−1
2π
∫ ∞−∞
w(x− iλ)
x− iλ− z dx. (37)
In particular, an application of the Cauchy-Schwarz inequality shows that for any y ∈ [−λ+ε, µ−ε]with ε > 0, w(z) is bounded in terms of the L2 norms of w(x+ iµ) and w(x− iλ).
Collecting these ideas, we choose the pressure p to be in the space of analytic functions in thesymmetric strip of width 2M about the real axis such that∫ ∞
−∞(1 + |x|2)|p(x+ iy)|2dx <∞, −M ≤ y ≤M. (38)
Note that this condition guarantees that the second condition (32) is also satisfied. Let HM denotethe space defined by (38). It is endowed with the norm
‖p‖HM = sup|y|≤M
[∫ ∞−∞
(1 + |x|2)|p(x+ iy)|2dx]1/2
. (39)
We claim that HM is a Banach space. Indeed, with the obvious definitions of addition and scalarmultiplication for elements p ∈ HM , HM is a vector space. It is straightforward to verify that (39)defines a norm. Thus, it remains to verify completeness. Let {fk} be a Cauchy sequence in HM .With the norm above, this sequence converges to a complex-valued function f defined on the stripsince for fixed y, {fk} defines a Cauchy sequence in the space L2 with weight (1 + |x|2) and has thelimit f(·, y) in this space for each y. We define the function
f(x, y) = limk→∞
fk(x+ iy), y fixed. (40)
Since {fk} is a Cauchy sequence, for every ε > 0 there exists an N such that for n, k ≥ N
‖fn − fk‖HM ≤ ε. (41)
This implies ∫ ∞−∞
(1 + |x|2)|fn(x+ iy)− fk(x+ iy)|2dx ≤ ε, (42)
for every |y| ≤M . Letting k →∞ in the above integral we obtain∫ ∞−∞
(1 + |x|2)|fn(x+ iy)− f(x, y)|2dx ≤ ε, (43)
10
for every |y| ≤M and thus fn → f in the HM norm. Theorem 3.2 implies the pointwise bound
|w(x+ iy)| ≤ C‖w(x+ iy)‖HM , (44)
for x+iy in the interior of the strip. Consequently, convergence in HM implies uniform convergenceon compact subsets of the strip. From Morera’s Theorem, it follows that the Cauchy sequence ofanalytic functions fk converges to an analytic function, thus the space HM is complete.
We now state the theorem for the existence of a map from the pressure beneath a travelingwave to the surface elevation of the wave.
Theorem 3.3 Let p and η be the bottom pressure and surface elevation, respectively, obtained bysolving the Euler equations augmented with (14). Assume that p ∈ HM+ε for some M > h, ε > 0and that ‖η‖C1 < min[M − h, c2/2g]. Then for fixed c 6= 0 and sufficiently small p, the equation
c−√c2 − 2gη
1 + η2x=
1
2π
∫ ∞−∞
eikx cosh(k(η + h))F{c−
√c2 − 2p
}(k)dk,
has a solution η. Further, if p is the true pressure consistent with (3-6,14), then the only solutionfor η is that which solves the stationary water wave problem with speed c.
Proof. Let M > h and ε > 0. By Theorem 3.2 there is a ball V around the origin in HM+ε,i.e. V = {p ∈ HM+ε : ‖p‖HM+ε
< δ}, such that
sup|y|≤M+ε/2
|p(x+ iy)| ≤ S < c2
2.
By definition, p ∈ V is bounded, analytic and square integrable along lines parallel to the real axis.Then the function c −
√c2 − 2p is also analytic and square integrable along lines parallel to the
real axis in a strip of width 2(M + ε/2) symmetric with respect to the real axis. Using Theorem3.1, ∫ ∞
−∞e2M |k|
∣∣∣F {c−√c2 − 2p}(k)∣∣∣2 dk <∞.
For p ∈ V , following the discussion preceeding Theorem 3.1, define the reconstructed functionsφR,x and φR,z as
φR,x(x, z; p) =1
2π
∫ ∞−∞
eikx cosh(k(z + h))F{c−
√c2 − 2p
}(k)dk, (45)
φR,z(x, z; p) =1
2π
∫ ∞−∞−ieikx sinh(k(z + h))F
{c−
√c2 − 2p
}(k)dk. (46)
Then φR,x and φR,z are harmonic (and thus smooth) for all x ∈ R and |z + h| < M . Let R =min
(M − h, c2/2g
)and define the ball U = {η ∈ C1 : ‖η‖C1 < R}. Then G : U × V → C1[R,R],
defined as
G(η, p) = −φR,x(x, η; p) +1
2c[φR,x(x, η; p)]2 +
1
2c[φR,z(x, η; p)]2 +
g
cη,
11
is a continuously differentiable function with G(0, 0) = 0, as is readily verified by computing itssecond variation evaluated at (η, p) = (0, 0). The Frechet derivative of G with respect to η at theorigin is
Gη(0, 0)v =g
cv, v ∈ C1.
The Frechet derivative Gη(0, 0) is an isomorphism on C1. Hence the Implicit Function Theoremapplies and there exists a continuously differentiable map ν : p→ η such that G(ν(p), p) = 0 for allsufficiently small p.
Next, we show that if p is the pressure consistent with the traveling water wave problem withvelocity c, then ν(p) is indeed the corresponding water wave surface elevation. This is achievedby establishing that the reconstructed functions φR,x and φR,z are the horizontal and vertical fluidvelocities φx and φz, respectively.
Let D = {−∞ < x <∞,−h < z < η}, as before, where η represents the solution for the surfaceelevation of the traveling water wave problem with velocity potential φ. Hence φ is harmonic in D.It is possible to harmonically extend φ to D = {−∞ < x <∞,−η − 2h < z < η} by reflecting theproblem across the mirror line z = −h. Thus φx is harmonic in D.
If p is the pressure corresponding to the solution (φ, η) of the water wave problem through(14), then at z = −h, φR,x and its normal derivative take the same values as φx and its normalderivative, respectively. The Cauchy-Kowalevski Theorem for Laplace’s equation [14] implies thatφx = φR,x in a region near z = −h. But then φx and φR,x and all their derivatives are equal inthis region. This implies that φx = φR,x in D, by analytic continuation. A similar argument showsthat φz = φR,z in D: to determine φzz at z = −h we use the fact that due to the extension to D,φ is harmonic on z = −h. In addition, from (45) and (46), φR,x and φR,z harmonically extend upto z = M − h > ‖η‖∞. Since φx = φR,x and φz = φR,z in D, we can harmonically extend φx andφz up to z = M − h. This implies that φR,x = φx and φR,z = φz at z = η. Hence, from (6) in thetraveling frame of reference,
−cφx +1
2φ2x +
1
2φ2z + gη = 0, z = η
⇒ −cφR,x +1
2(φR,x)2 +
1
2(φR,z)2 + gη = 0, z = η,
⇒ G(η, p) = 0.
From the Implicit Function Theorem, for small p all solutions η to G(η, p) = 0 are of the formη = ν(p), where ν : p→ η is a C1 map. Hence
φR,x(x, ν(p); p) = φR,x(x, η; p) = φx(x, η) = c−√c2 − 2gη
1 + η2x= c−
√c2 − 2gν(p)
1 + ν2x(p),
where we used (13). In other words, there are functions η which depend continuously on the truepressure p such that (21) is true.
Next, assume there exists a different solution η ∈ U ⊂ C1, η 6= η such that
c−√c2 − 2gη
1 + η2x= φR,x(x, η; p),
12
for all x, where the pressure p is the pressure corresponding to the traveling water wave problemwith velocity c. As before, φR,x(x, η; p) = φx(x, η) and thus
c−√c2 − 2gη
1 + η2x= φx(x, η),
for all x. However, for a fixed c, the water wave problem has a unique traveling wave solution [3],which is contradicted by the statement that η 6= η. Thus the only solutions η of (21) associatedwith the pressure p are the traveling wave solutions of the Euler equations.
4 Asymptotic approximations
In this section we derive a variety of asymptotic approximations to the pressure as a function of thesurface elevation. Given the complexity of (21), such approximations are especially useful. In thesections below, we compare the results for the pressure obtained using (21), with those obtainedfrom (1) and (2), as well as some asymptotic formulas obtained here.
We introduce the nondimensional quantities ξ∗, z∗, η∗ and k∗:
ξ∗ = ξ/L, z∗ = z/h, η∗ = η/a, k∗ = Lk, c∗ = c/√gh, p∗ = p/gh, (47)
where L is a typical horizontal length scale, and a is the amplitude of the surface wave. From (3–6),a nondimensional version of (21) is found to be√
c2 − 2εη
1 + (εµηξ)2=
1
2π
∫ ∞−∞
eikξF{√
c2 − 2εp(ξ)}
(k) cosh (µk (1 + εη)) dk, (48)
where ε = a/h and µ = h/L. The ∗’s have been omitted to simplify the notation. This form of thenonlocal relation is our starting point to derive various approximate results. The two parameters εand µ provide many options for different asymptotic expansions: we may assume small amplitudewaves (ε � µ), or we may assume a long-wave approximation (µ � ε), or we may balance botheffects as in a Korteweg-de Vries (KdV)-type approximation (see [2], for instance).
4.1 The small-amplitude approximation: SAO1 and SAO2
If we expand η in powers of ε� 1, assuming that ε� µ, we recover at leading order the approxi-mation
η(ξ) = F−1{
cosh(µk)p(k)}}
+O (ε) , (49)
where p(k) = F{p}(k). Ignoring the O (ε) term, (49) is the nondimensional version of (2). Thisdemonstrates that (21) is consistent with the frequently used (2), and we are able to recover suchformulas in a consistent manner using the single equation (21), instead of having to work with thefull set of equations of motion. For the remainder of this paper, we will refer to the model (49)(solved for the pressure) as SAO1 (Small-Amplitude, Order 1).
If we proceed to higher order in ε we find the presumably more accurate approximation
η(ξ) = η0(ξ) + εη1(ξ) +O(ε2), (50)
13
where
η0(ξ) =F−1{
cosh(µk)p(k)}}, (51)
η1(ξ) =− c2µ2
2η0
2ξ −
1
2c2η20 + µη0F−1{kp(k) sinh(µk)}+
1
2c2F−1{p2(k) cosh(µk)}. (52)
The formula (50) provides a new, explicit, higher-order approximation for the surface elevationη(ξ) in terms of the pressure p(ξ) and the traveling wave speed c, assuming a small-amplitudeapproximation. For the remainder of this paper, we will refer to this model as SAO2.
4.2 The KdV Approximation: SWO1 and SWO3
Alternatively, we can balance the parameters µ and ε so that µ =√ε. This is the KdV approxima-
tion, see [1, 2]). At leading order, we recover the simplest approximation that the surface elevationequals the pressure:
η(ξ) = p(ξ) +O(ε). (53)
This equation is exactly the hydrostatic approximation (1) in dimensionless variables; we willrefer to this model as SWO1 (Shallow Water, Order 1). Continuing the aproximations to higherorder (up to order ε3), we find
η(ξ) = p− ε
2
∂2p
∂ξ2+ ε2
(1
24
∂4p
∂ξ4−p∂
2p
∂ξ2− 1
2
(∂p
∂ξ
)2(c2 +
1
c2
))+O(ε3). (54)
We refer to this approximation as SWO3 (Shallow Water, Order 3).
Remarks.
• For η ∈ C1 and p ∈ HM+δ, it is possible to prove the analyticity of (48) in ε and µ. Here, asbefore, M is related to the size of a symmetric strip around the real ξ axis. This analyticityserves to validate the asymptotic approximations derived above, as being obtained through aprocess that gives the first few terms of a convergent series.
• It appears to be a big restriction that the nonlocal formula (21) and the approximationsderived above require a traveling wave profile. As shown in the next section, good resultsare also obtained for waves that are not merely traveling at constant speed. For waves inshallow water, excellent agreement is often obtained by using c = 1 (or c =
√gh, returning
to the dimensional version), which may be regarded as the zero-order approximation of anasymptotic series for c in terms of ε.
• Using the procedures outlined in this section, the reader will find it straightforward to deriveyet different approximations for the surface elevation in terms of the pressure measured atthe bottom. For instance, one may consider a shallow-water approximation without imposingthat the waves are of small amplitude, i.e., µ� ε < 1, etc.
14
4.3 A heuristic formula: SAO2h
The transfer function approach (2) is very successful for a variety of reasons: (i) it is quite accurate,as is illustrated in the next few sections. This statement remains true to a varying degree for waves ofrelatively high amplitude; (ii) the most complicated aspect of using the formula is the computationof two Fourier transforms; and (iii) the formula applies to waves that are not necessarily travelingwith constant speed. This is a consequence of the linearization that led to (2): each individuallinear wave is traveling at constant speed, but typically their superposition is not.
In this section we derive a different formula for the reconstruction of the surface elevation fromthe pressure at the bottom. This formula is obtained somewhat heuristically, and its justificationrests on the fact that it agrees extremely well with both numerical and experimental data. Fur-thermore, its use requires the computation of only three Fourier transforms, and the velocity c doesnot appear in the final result. As a consequence, even though the derivation does not justify this,it is straightforward to apply to non-traveling wave profiles, where it performs very well. As forthe other formulas above, the numerical and experimental results are presented below.
An equivalent form of the nondimensional nonlocal equation (48) is
1−√
1− 2εη/c2
1 + (εµηξ/c)2=
1
2π
∫ ∞−∞
eikξP (k, ε) cosh (µk (1 + εη)) dk, (55)
where
P (k, ε) = F{
1−√
1− 2εp(ξ)/c2}
(k). (56)
So as to consider a small-amplitude approximation, we expand this equation in powers of ε. How-ever, we do not expand P (k, ε) at this point. Proceeding this way and retaining only first-orderterms in εη and εηξ, we find
εη
c2=
1
2π
∫ ∞−∞
eikξP (k, ε) (cosh(µk) + εµηk sinh(µk)) dk
⇒ εη =12π
∫∞−∞ e
ikξP (k, ε) cosh(µk) dk
1c2− µ
2π
∫∞−∞ e
ikξP (k, ε)k sinh(µk) dk. (57)
Next, we expand P (k, ε) in ε, omitting terms of order ε2 and higher. We obtain
P (k, ε) =ε
c2p. (58)
Substitution of (58) in (57) results in
η =F−1 {p(k) cosh(µk)}
1− εµF−1 {p(k) k sinh(µk)} . (59)
As stated above, this reconstruction formula does not depend on c, and its application requiresthe computation of a mere three Fourier transforms. This can be contrasted, for instance, withthe formula SAO2 which also uses a small-amplitude approximation. That formula requires thecomputation of five Fourier transforms and has explicit dependence on c. In fact, if one were toexpand (59) in powers of ε one would find at order ε0 the transfer function formula (2), and at
15
order ε1 the result SAO2 with all c-dependent terms omitted. We refer to the results obtainedusing (59) as SAO2h.
5 Comparisons of the Different Approaches
In this section, we present numerical results for the reconstruction of the surface elevation usingthe various relationships derived in Section 4 for both numerical and experimental pressure data.For the comparison using numerical data, we use previously computed periodic traveling wavessolutions from [11]. By using the exact pressure underneath the traveling wave, we attempt toreconstruct the surface elevation. The same is done for various sets of experimental data obtainedfrom the one-dimensional wave tank at the William Pritchard Fluids Laboratory at Penn StateUniversity.
5.1 Comparison of the Different Approaches Using Numerical Data
Using traveling wave solutions with periodic boundary conditions as calculated in [11], we determinethe pressure at the bottom using (21) as follows. Without loss of generality, we assume that thesolutions are 2π periodic. For a given traveling wave solution profile specified by (ηtrue(ξ), ctrue),the pressure p(ξ) is obtained by equating the k-th Fourier coefficient of both the right- and left-handside of (21) for k = −N . . .N , using a sufficiently high value of N . This results in a linear systemof algebraic equations for the coefficients of the Fourier series of
√c2 − 2p. Using this truncated
Fourier series, we may solve directly for p(ξ) in terms of the given solution set (ηtrue(ξ), ctrue). Notethat (25) offers a numerically equivalent alternative for computing p(ξ).
Our goal is to reconstruct the surface elevation from the thus computed pressure at the bottom,using the various formulas given above. The asymptotic formulas given in the previous section donot require anything more complicated than a fast Fourier transform. The solution of the nonlocalequation (21) is obtained using a pseudo-spectral method with differentiation carried out in Fourierspace, while multiplication is carried out in physical space. We reconstruct η by using a nonlinearsolver such as a Gauss-Newton or Dogleg method [12, 22] with an error tolerance of 10−14. As aninitial guess for our nonlinear solver, we use the approximation from (50). Of course, the resultobtained from the nonlocal equation should return the original surface elevation profile used togenerate the pressure data, within machine precision. This provides a validation for the variousnumerical methods used. Here we compare the results from the asymptotic formulas of the previoussection and evaluate their different errors.
Using the parameter values h = .1, g = 1, ρ = 1 and L = 2π, we reconstruct the solution forvarious solution amplitudes and speeds. For solutions of small amplitude (say ak = .0001), we seethat the reconstructions using all methods are in excellent agreement with the true surface waveelevation, see Figure 2a. However, even for waves with amplitudes less than 15% of the limitingwave height as given by [7] it becomes clear that certain approximations yield better results thanothers, see Figure 2b. In particular, while the nonlocal formula and the higher-order asymptoticformula reconstruct the wave profile well, the hydrostatic approximation SWO1 reconstruction failsto reproduce an accurate reconstruction of the peak wave height.
To demonstrate how the error changes as a function of the wave amplitude (or nonlinearity),we compute the relative error
16
2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 44
5
6
7
8
9
10
11x 10 5
Surfa
ce E
leva
tion
()
(a)
2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 42
0
2
4
6
8
10
12x 10 3
Surfa
ce E
leva
tion
()
(b)
Figure 2: Reconstruction of the surface elevation from pressure data based on numerical experimentsfor h = 0.1, g = 1, ρ = 1 and L = 2π. Amplitudes are a = 0.0001 (a) and a = 0.0056. (b). Nolegend is included: all approximations including the nonlocal formula (21) result in indistinguishablecurves, except for the hydrostatic approximation SWO1, which displays a significant discrepancyfor the bottom numerical experiment.
error =||ηtrue − ηr||∞||ηtrue||∞
, (60)
where ηtrue represents the expected solution and ηr represents the reconstructed solution. For thesame nondimensional parameters as before, we calculate the error as a function of increasing peakwave height demonstrated in Figure 3. As seen there, the error in all approximations grows as theamplitude of the Stokes wave increases. Figure 3 includes only solutions of small amplitude. Ifsolutions of larger amplitude are considered, the discrepancies between the different approximationsgrow, as shown in Table 4.
This table illustrates the large error generated by the lower-order methods SAO1 and SW01 forwaves which are no more than 55% of the limiting wave height as calculated in [7]. Even for waveswhich are 50% of the limiting wave height, the relative error (60) of the commonly used transferfunction reconstruction SAO1 exceeds 15%. In contrast, the higher-order methods SA02, SWO3,and SA02h consistently yield more accurate results. It is also clear from the table that even forlarge amplitude waves, the nonlocal formula (21) (or more precisely for these numerical data sets,its periodic analogue (22)) provides a practical means to reconstruct the surface elevation frompressure data measured along the bottom of a fluid, at least in this numerical data setting. Belowwe establish the same using physical experiments.
One limitation of the nonlocal equation (21) and some of its asymptotic counterparts from the
17
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5x 10 3
10 12
10 10
10 8
10 6
10 4
10 2
Amplitude
Erro
rCalculated using ctrue
Nonlocal EquationSWO3SAO1SAO2h
Figure 3: Plot of the relative error (60) in the reconstructed surface elevation ηr as a function ofthe amplitude of ηtrue using the true value of the wave speed c. The asymptotic approximationSAO2 is not included in this figure. It is more costly to compute than its heuristic counterpartSAO2h, which yields better results.
previous section is that they require the knowledge of the traveling wave speed c. In practice, thiscan be a difficult or impractical quantity to measure. One such impractical option is to includeadditional pressure sensors in order to measure the time it takes for the peak of the pressure datato travel from one sensor to another. A simpler option is to use approximations for the wave speedbased on small-amplitude theory. For example, if we repeat the same error calculation as above,but with c ≈ √gh, we obtain Figure 5. We might hope that the reconstruction of the surfaceelevation would not suffer much. In fact, it appears unchanged. As seen in Figure 5, the error inreconstructing the peak wave height does not suffer at all from using this simple approximate (andamplitude-independent) value of c. Surprisingly, the error in the nonlocal reconstruction remainsconsistent with the error calculated using the true wave speed c. This lack of sensitivity to theprecise value of c yields hope that with experimental data a simple approximation of the wave speedwill be sufficient to accurately reconstruct the surface elevation.
Percentage ofLimiting Wave Height SWO1 SAO1 SWO3 SAO2 SAO2h Nonlocal
35 21.14 9.43 4.01 2.36 1.76 0.0045 25.67 13.43 6.49 4.27 3.18 0.0050 27.88 15.49 7.89 5.41 4.04 0.0053 28.94 16.49 8.67 5.96 4.40 0.0054 29.65 12.17 9.15 6.36 4.70 0.0055 30.08 17.58 9.45 6.61 4.88 0.00
Table 4: Relative error (60) in percent, calculated comparing peak wave heights using variousreconstruction formulas using numerical data.
18
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5x 10 3
10 14
10 12
10 10
10 8
10 6
10 4
10 2
Amplitude
Erro
rCalculated using capprox
Nonlocal EquationSWO3SAO1SAO2h
Figure 5: Plot of the error in the reconstructed surface elevation ηr as a function of the amplitudeof ηtrue using an approximation for the wave-speed c. The SAO2 model is not included for thesame reason as in Figure 3.
5.2 Comparison of the Different Approaches Using Data From Physical Exper-iments
Here we discuss comparisons of results from the nonlocal formula (21) and the asymptotic ap-proaches with results from ten laboratory experiments performed at Penn State’s Pritchard FluidMechanics Laboratory. In these experiments the pressure at the bottom of the fluid domain andthe displacement of the air–water interface were measured simultaneously. The experimental fa-cility consisted of the wave channel and water, the wavemaker, bottom pressure transducers, anda surface displacement measurement system. The wavetank is 50 ft long, 10 in wide and 1 ft deep.It is constructed of tempered glass. It was filled with tap water to a depth of h, as listed inTable 8. The pressure gage was a SENZORS PL6T submersible level transducer with a range of0–4 in. It provided a 0–5 V dc output, which was digitized with an NI PCI-6229 analog-to-digitalconverter using LabView software. We calibrated this transducer by raising and lowering the waterlevel in the channel. The pressure measurements had a high-frequency noise component, and thuswere low-pass filtered at 20 Hz. The still-water height was measured with a Lory Type C pointgage. The capacitance-type surface wave gage consisted of a coated-wire probe connected to anoscillator. The difference frequency between this oscillator and a fixed oscillator was read by aField Programmable Gate Array (FPGA), NI PCI-7833R. Thus, no D/A conversion, filtering, orA/D conversion was required. The surface capacitance gage was held in a rack on wheels that areattached to a programmable belt. We calibrated the capacitance gage by traversing the rack ata known speed over a precisely machined, trapezoidally-shaped “speed bump”. The waves werecreated with a horizontal, piston-like motion of a paddle made from a Teflon plate (0.5 in thick)inserted in the channel cross-section. The paddle was machined to fit the channel precisely witha thin lip around its periphery that served as a wiper with the channel’s glass perimeter. Thiswiper prevented any measurable leakage around the paddle during an experiment. The paddle was
19
15 20 25 30 35 40 45 50 55 60 650.4
0.2
0
0.2
0.4
0.6
0.8
1
Surfa
ce E
leva
tion
(
)
True SolutionSWO1SWO3SAO1SAO2hNonlocal Equation
Figure 6: Wave tank comparisons of the reconstructed surface elevation with surface height mea-surements for h = 5.05 cm. This corresponds to Experiment #1 in Table 8.
connected to the programmable belt and traveled in one direction. It was programmed with thehorizontal velocity of a KdV soliton, which is given by
u(x, t) = u0 sech2
(3u0
4h02c0
(x− c0t− u0t/2)
), (61)
where c0 =√gh, a0 is the wave amplitude, and u0 = a0c0/h is the maximum horizontal velocity.
The a0 for the wavemaker displacement was varied between 2 cm and 3 cm. These values corre-sponded to large velocities and fluid displacements, outside of the regime of the KdV equation. Thewater adjusted to create a leading wave with a radiative tail. We compare results for the leadingwave, where nonlinearity is likely to be important.
To convert the time series of pressure and surface displacement data into spatial data, weuse a combination of the sampling frequency and the estimated wave speed c. Specifically, letpj represents the measured pressure at time tj = j∆t, where ∆t is the time between pressuremeasurements. We assign a corresponding x value xj to pj so that xj = c (j∆t). From thepressure data measured from the physical experiments, we reconstruct the surface elevation usingthe same methods as in the previous section. For all experiments we use the admittedly simpleapproximation c ≈ √gh. We use the measured pressure data to reconstruct the surface elevationusing the nonlocal formula (21), as well as the asymptotic approximations SWO1 (hydrostatic),SAO1 (transfer function), SAO2, and SWO3.
As seen in Figure 6, the higher-order methods capture the peak wave height better than thelower-order methods, with the nonlocal equation yielding the most accurate representation of thepeak wave height. The visual comparisons for all experiments is displayed in Figure 7. The nonlocalformula (21) consistently captures the peak wave height better than any of the approximate modelsderived in the previous section, and significantly better than the SWO1 (hydrostatic) and SAO1(transfer function) models. This is quantified in Table 8, which displays the error (60) for thedifferent approximations and the nonlocal equation (21). As is seen there, the result from (21)consistently produced the smallest error among all the reconstruction formulas. It is noteworthy
20
0 20 40 60 800.5
0
0.5
1
0 20 40 60 800.5
0
0.5
1
0 20 40 60 800.5
0
0.5
1
0 20 40 60 800.5
0
0.5
1
0 20 40 60 800.5
0
0.5
1
0 20 40 600.5
0
0.5
1
0 20 40 600.5
0
0.5
1
0 20 40 600.5
0
0.5
1
0 20 40 600.5
0
0.5
1
0 20 40 600.5
0
0.5
1
Figure 7: Wave tank comparisons of the reconstructed surface elevation with surface height mea-surements for various fluid depths. The experiments are ordered from left-to-right and top-to-bottom, corresponding to the experiment # in Table 8.
that the heuristic SAO2h approximation (59) consistently yields the second-lowest peak heighterror and consistently outperforms all other models except the nonlocal equation (21). Given thecomputational expense of solving the nonlocal equation, the approximation SAO2h apparentlyyields the best compromise between efficiency and accuracy.
6 Conclusion
We have presented a new equation (21) relating the pressure at the bottom of the fluid to thesurface elevation of a traveling wave solution of the one-dimensional Euler equations without ap-proximation. This equation is analyzed rigorously and the existence of solutions is proven using theImplicit Function Theorem. Solving the equation numerically is possible, but this is computation-ally relatively expensive when compared to currently-used approaches that require the computationof at most a few Fourier transforms. To this end, we derive various new approximate formulas,starting from the new nonlocal formula. The canonical approaches (hydrostatic approximation andtransfer function approach) are easily obtained from the nonlocal formula as well.
The different approximations and the nonlocal formula (21) are compared using numerical data,and their performance on physical laboratory data is examined. The nonlocal formula consistentlyoutperforms its different approximations. For the numerical data this is by construction, as it wasused to generate the numerical pressure data used for the comparison, starting from computedtraveling wave solutions of the Euler equation. The higher-order approximate formulas result in abetter reconstruction of the surface elevation compared to the hydrostatic or transfer function ap-
21
Experiment # Depth (cm) SWO1 SAO1 SWO3 SAO2 SAO2h Nonlocal
1 5.05 22.29 6.51 2.35 0.84 0.45 0.202 5.05 24.66 8.05 3.30 1.13 0.56 0.013 5.05 23.56 7.75 2.90 0.95 0.41 0.184 5.05 21.98 7.18 2.80 1.29 0.89 0.665 5.05 22.00 6.63 2.10 0.48 0.05 0.036 3.55 18.11 5.21 1.65 0.65 0.43 0.367 3.55 20.81 7.04 3.33 2.22 1.95 1.528 4.10 21.59 8.39 3.50 2.18 1.73 0.939 4.10 22.13 7.65 2.29 0.33 0.31 0.0510 4.10 23.32 9.25 4.60 2.91 2.44 2.13
Table 8: Relative error (60) in percent, calculated comparing peak wave heights using variousreconstruction formulas using experimental data.
proaches. In the lab experiments, both the surface elevation and the bottom pressure are measured,allowing for an independent validation of the nonlocal equation (21). As expected, it outperformsthe different approximations, where higher-order models perform better than lower-order ones. Agood compromise between computational cost and obtained accuracy seems to be achieved by theheuristic approximation SAO2h (59), which requires the computation of three Fourier transforms.
Our derivation of the nonlocal equation (21) requires the surface elevation profile to be travelingwith constant speed c. Regardless, we show that the results are not sensitive to the exact value ofc and even rough estimates (i.e., c =
√gh) provide excellent results, both for the nonlocal equation
(21) and its various asymptotic approximations, most notably (59).
Acknowledgements
BD and VV acknowledge support from the National Science Foundation under grant NSF-DMS-1008001. DH acknowledges support from the National Science Foundation under grants NSF-DMS-0708352 and NSF-DMS-1107379. She is grateful to Rod Kreuter and Rob Geist for development ofthe electrical and mechanical systems for the experiments. Any opinions, findings, and conclusionsor recommendations expressed in this material are those of the authors and do not necessarilyreflect the views of the funding sources.
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